Semiclassical instabilities of Kerr-anti-de Sitter black holes

We study the thermodynamic stability of the Kerr-anti-de Sitter black hole from the perturbative corrections to the gravitational partition function. The line of critical stability is identified by the appearance of a negative mode of the Euclidean action that renders the partition function ill defined. The eigenvalue problem, consisting of a system of three coupled partial differential equations for the metric perturbations, is solved numerically. The agreement with the standard condition of thermodynamic stability in the grand-canonical ensemble is remarkable. The results illustrate the physical significance of gravitational partition functions for rotating spacetimes beyond the instanton approximation. At a classical level, the results also imply that the Gregory-Laflamme instability of the Kerr string persists up to extremality, the range of unstable modes increasing with the angular momentum.